ACT Math: Inverse Trig Functions—Domain, Range, and Applications

Regular trig functions (sin, cos, tan) are not one-to-one; they repeat values. To define inverse functions (sin⁻¹, cos⁻¹, tan⁻¹), we restrict the domain. sin⁻¹ has domain [-1,1] and range [-π/2, π/2]. cos⁻¹ has domain [-1,1] and range [0, π]. tan⁻¹ has domain (-∞, ∞) and range (-π/2, π/2). The restricted range means sin⁻¹(x) returns values only in [-π/2, π/2], not all possible angles whose sine is x. This is crucial: if you want all angles whose sine is 0.5, you must account for the restriction and find complementary angles manually.

Example: sin⁻¹(0.5)=π/6 (within the restricted range). But sin(π/6)=sin(5π/6)=0.5. The inverse function returns only π/6.

Calculation 1: Find θ where sin(θ)=0.866 and 0<θ<π. θ=sin⁻¹(0.866)=π/3 (60°). Calculation 2: Find θ where cos(θ)=-0.5 and 0<θ<π. θ=cos⁻¹(-0.5)=2π/3 (120°). Calculation 3: Find θ where tan(θ)=1 and -π/2<θ<π/2. θ=tan⁻¹(1)=π/4 (45°). Remember the range restrictions. If a question asks for all angles in [0, 2π), you must find both the inverse trig value and its supplement/reference angle counterpart.

Practice inverse trig calculations daily until you quickly compute sin⁻¹, cos⁻¹, tan⁻¹ values.

Problem 1: Find θ in [-π/2, π/2] where sin(θ)=0.5. θ=sin⁻¹(0.5)=π/6. Problem 2: Find θ in [0, π] where cos(θ)=0. θ=cos⁻¹(0)=π/2. Problem 3: Find θ in (-π/2, π/2) where tan(θ)=-1. θ=tan⁻¹(-1)=-π/4. Problem 4: Find all θ in [0, 2π) where sin(θ)=0.5. Inverse trig gives π/6. Reference angle also gives 5π/6 (π-π/6). Answers: π/6 and 5π/6. Complete all four daily until you confidently use inverse trig and find reference angles.

Verify: sin(π/6)=0.5 ✓. sin(5π/6)=0.5 ✓.

Inverse trig questions appear in 1-2 ACT Math sections, usually medium-to-hard difficulty. They reward understanding domain/range restrictions over memorization. A student who understands why inverse functions need restricted domains can answer these confidently; one who doesn't must guess. This knowledge gap is worth 1-2 points.

Master inverse trig in one study session. By test day, restricted domains and inverse function calculations become automatic.